direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4×Dic10, C42.271D10, C10⋊2(C4×Q8), C20⋊6(C2×Q8), (C2×C20)⋊14Q8, (C2×C42).18D5, C10.1(C22×Q8), (C2×C10).11C24, C10.22(C23×C4), C20.176(C22×C4), (C4×C20).332C22, (C2×C20).873C23, (C22×C4).465D10, Dic5.9(C22×C4), C2.1(C22×Dic10), C22.11(C23×D5), C22.65(C4○D20), C4⋊Dic5.394C22, C22.33(C2×Dic10), C23.309(C22×D5), (C22×C10).373C23, (C22×C20).500C22, (C2×Dic5).181C23, (C4×Dic5).329C22, (C22×Dic10).21C2, (C2×Dic10).321C22, C10.D4.173C22, (C22×Dic5).222C22, C5⋊2(C2×C4×Q8), C4.75(C2×C4×D5), (C2×C4×C20).22C2, C2.4(D5×C22×C4), C10.1(C2×C4○D4), C2.1(C2×C4○D20), C22.66(C2×C4×D5), (C2×C4).118(C4×D5), (C2×C10).45(C2×Q8), (C2×C4×Dic5).43C2, (C2×C20).403(C2×C4), (C2×C4⋊Dic5).48C2, (C2×C10).93(C4○D4), (C2×C4).815(C22×D5), (C2×C10).245(C22×C4), (C2×Dic5).114(C2×C4), (C2×C10.D4).38C2, SmallGroup(320,1139)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 750 in 298 conjugacy classes, 183 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×14], C22, C22 [×6], C5, C2×C4 [×14], C2×C4 [×22], Q8 [×16], C23, C10 [×3], C10 [×4], C42 [×4], C42 [×8], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×12], Dic5 [×8], Dic5 [×4], C20 [×8], C20 [×2], C2×C10, C2×C10 [×6], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, Dic10 [×16], C2×Dic5 [×16], C2×Dic5 [×4], C2×C20 [×14], C2×C20 [×2], C22×C10, C2×C4×Q8, C4×Dic5 [×8], C10.D4 [×8], C4⋊Dic5 [×4], C4×C20 [×4], C2×Dic10 [×12], C22×Dic5 [×4], C22×C20 [×3], C4×Dic10 [×8], C2×C4×Dic5 [×2], C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C4×C20, C22×Dic10, C2×C4×Dic10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], D5, C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, Dic10 [×4], C4×D5 [×4], C22×D5 [×7], C2×C4×Q8, C2×Dic10 [×6], C2×C4×D5 [×6], C4○D20 [×2], C23×D5, C4×Dic10 [×4], C22×Dic10, D5×C22×C4, C2×C4○D20, C2×C4×Dic10
Generators and relations
G = < a,b,c,d | a2=b4=c20=1, d2=c10, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 320 points
Generators in S320
(1 182)(2 183)(3 184)(4 185)(5 186)(6 187)(7 188)(8 189)(9 190)(10 191)(11 192)(12 193)(13 194)(14 195)(15 196)(16 197)(17 198)(18 199)(19 200)(20 181)(21 217)(22 218)(23 219)(24 220)(25 201)(26 202)(27 203)(28 204)(29 205)(30 206)(31 207)(32 208)(33 209)(34 210)(35 211)(36 212)(37 213)(38 214)(39 215)(40 216)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 225)(62 226)(63 227)(64 228)(65 229)(66 230)(67 231)(68 232)(69 233)(70 234)(71 235)(72 236)(73 237)(74 238)(75 239)(76 240)(77 221)(78 222)(79 223)(80 224)(81 173)(82 174)(83 175)(84 176)(85 177)(86 178)(87 179)(88 180)(89 161)(90 162)(91 163)(92 164)(93 165)(94 166)(95 167)(96 168)(97 169)(98 170)(99 171)(100 172)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(241 269)(242 270)(243 271)(244 272)(245 273)(246 274)(247 275)(248 276)(249 277)(250 278)(251 279)(252 280)(253 261)(254 262)(255 263)(256 264)(257 265)(258 266)(259 267)(260 268)(281 301)(282 302)(283 303)(284 304)(285 305)(286 306)(287 307)(288 308)(289 309)(290 310)(291 311)(292 312)(293 313)(294 314)(295 315)(296 316)(297 317)(298 318)(299 319)(300 320)
(1 214 48 285)(2 215 49 286)(3 216 50 287)(4 217 51 288)(5 218 52 289)(6 219 53 290)(7 220 54 291)(8 201 55 292)(9 202 56 293)(10 203 57 294)(11 204 58 295)(12 205 59 296)(13 206 60 297)(14 207 41 298)(15 208 42 299)(16 209 43 300)(17 210 44 281)(18 211 45 282)(19 212 46 283)(20 213 47 284)(21 160 308 185)(22 141 309 186)(23 142 310 187)(24 143 311 188)(25 144 312 189)(26 145 313 190)(27 146 314 191)(28 147 315 192)(29 148 316 193)(30 149 317 194)(31 150 318 195)(32 151 319 196)(33 152 320 197)(34 153 301 198)(35 154 302 199)(36 155 303 200)(37 156 304 181)(38 157 305 182)(39 158 306 183)(40 159 307 184)(61 279 177 107)(62 280 178 108)(63 261 179 109)(64 262 180 110)(65 263 161 111)(66 264 162 112)(67 265 163 113)(68 266 164 114)(69 267 165 115)(70 268 166 116)(71 269 167 117)(72 270 168 118)(73 271 169 119)(74 272 170 120)(75 273 171 101)(76 274 172 102)(77 275 173 103)(78 276 174 104)(79 277 175 105)(80 278 176 106)(81 137 221 247)(82 138 222 248)(83 139 223 249)(84 140 224 250)(85 121 225 251)(86 122 226 252)(87 123 227 253)(88 124 228 254)(89 125 229 255)(90 126 230 256)(91 127 231 257)(92 128 232 258)(93 129 233 259)(94 130 234 260)(95 131 235 241)(96 132 236 242)(97 133 237 243)(98 134 238 244)(99 135 239 245)(100 136 240 246)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260)(261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)
(1 273 11 263)(2 272 12 262)(3 271 13 261)(4 270 14 280)(5 269 15 279)(6 268 16 278)(7 267 17 277)(8 266 18 276)(9 265 19 275)(10 264 20 274)(21 96 31 86)(22 95 32 85)(23 94 33 84)(24 93 34 83)(25 92 35 82)(26 91 36 81)(27 90 37 100)(28 89 38 99)(29 88 39 98)(30 87 40 97)(41 108 51 118)(42 107 52 117)(43 106 53 116)(44 105 54 115)(45 104 55 114)(46 103 56 113)(47 102 57 112)(48 101 58 111)(49 120 59 110)(50 119 60 109)(61 289 71 299)(62 288 72 298)(63 287 73 297)(64 286 74 296)(65 285 75 295)(66 284 76 294)(67 283 77 293)(68 282 78 292)(69 281 79 291)(70 300 80 290)(121 141 131 151)(122 160 132 150)(123 159 133 149)(124 158 134 148)(125 157 135 147)(126 156 136 146)(127 155 137 145)(128 154 138 144)(129 153 139 143)(130 152 140 142)(161 214 171 204)(162 213 172 203)(163 212 173 202)(164 211 174 201)(165 210 175 220)(166 209 176 219)(167 208 177 218)(168 207 178 217)(169 206 179 216)(170 205 180 215)(181 246 191 256)(182 245 192 255)(183 244 193 254)(184 243 194 253)(185 242 195 252)(186 241 196 251)(187 260 197 250)(188 259 198 249)(189 258 199 248)(190 257 200 247)(221 313 231 303)(222 312 232 302)(223 311 233 301)(224 310 234 320)(225 309 235 319)(226 308 236 318)(227 307 237 317)(228 306 238 316)(229 305 239 315)(230 304 240 314)
G:=sub<Sym(320)| (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,196)(16,197)(17,198)(18,199)(19,200)(20,181)(21,217)(22,218)(23,219)(24,220)(25,201)(26,202)(27,203)(28,204)(29,205)(30,206)(31,207)(32,208)(33,209)(34,210)(35,211)(36,212)(37,213)(38,214)(39,215)(40,216)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,225)(62,226)(63,227)(64,228)(65,229)(66,230)(67,231)(68,232)(69,233)(70,234)(71,235)(72,236)(73,237)(74,238)(75,239)(76,240)(77,221)(78,222)(79,223)(80,224)(81,173)(82,174)(83,175)(84,176)(85,177)(86,178)(87,179)(88,180)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(241,269)(242,270)(243,271)(244,272)(245,273)(246,274)(247,275)(248,276)(249,277)(250,278)(251,279)(252,280)(253,261)(254,262)(255,263)(256,264)(257,265)(258,266)(259,267)(260,268)(281,301)(282,302)(283,303)(284,304)(285,305)(286,306)(287,307)(288,308)(289,309)(290,310)(291,311)(292,312)(293,313)(294,314)(295,315)(296,316)(297,317)(298,318)(299,319)(300,320), (1,214,48,285)(2,215,49,286)(3,216,50,287)(4,217,51,288)(5,218,52,289)(6,219,53,290)(7,220,54,291)(8,201,55,292)(9,202,56,293)(10,203,57,294)(11,204,58,295)(12,205,59,296)(13,206,60,297)(14,207,41,298)(15,208,42,299)(16,209,43,300)(17,210,44,281)(18,211,45,282)(19,212,46,283)(20,213,47,284)(21,160,308,185)(22,141,309,186)(23,142,310,187)(24,143,311,188)(25,144,312,189)(26,145,313,190)(27,146,314,191)(28,147,315,192)(29,148,316,193)(30,149,317,194)(31,150,318,195)(32,151,319,196)(33,152,320,197)(34,153,301,198)(35,154,302,199)(36,155,303,200)(37,156,304,181)(38,157,305,182)(39,158,306,183)(40,159,307,184)(61,279,177,107)(62,280,178,108)(63,261,179,109)(64,262,180,110)(65,263,161,111)(66,264,162,112)(67,265,163,113)(68,266,164,114)(69,267,165,115)(70,268,166,116)(71,269,167,117)(72,270,168,118)(73,271,169,119)(74,272,170,120)(75,273,171,101)(76,274,172,102)(77,275,173,103)(78,276,174,104)(79,277,175,105)(80,278,176,106)(81,137,221,247)(82,138,222,248)(83,139,223,249)(84,140,224,250)(85,121,225,251)(86,122,226,252)(87,123,227,253)(88,124,228,254)(89,125,229,255)(90,126,230,256)(91,127,231,257)(92,128,232,258)(93,129,233,259)(94,130,234,260)(95,131,235,241)(96,132,236,242)(97,133,237,243)(98,134,238,244)(99,135,239,245)(100,136,240,246), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,273,11,263)(2,272,12,262)(3,271,13,261)(4,270,14,280)(5,269,15,279)(6,268,16,278)(7,267,17,277)(8,266,18,276)(9,265,19,275)(10,264,20,274)(21,96,31,86)(22,95,32,85)(23,94,33,84)(24,93,34,83)(25,92,35,82)(26,91,36,81)(27,90,37,100)(28,89,38,99)(29,88,39,98)(30,87,40,97)(41,108,51,118)(42,107,52,117)(43,106,53,116)(44,105,54,115)(45,104,55,114)(46,103,56,113)(47,102,57,112)(48,101,58,111)(49,120,59,110)(50,119,60,109)(61,289,71,299)(62,288,72,298)(63,287,73,297)(64,286,74,296)(65,285,75,295)(66,284,76,294)(67,283,77,293)(68,282,78,292)(69,281,79,291)(70,300,80,290)(121,141,131,151)(122,160,132,150)(123,159,133,149)(124,158,134,148)(125,157,135,147)(126,156,136,146)(127,155,137,145)(128,154,138,144)(129,153,139,143)(130,152,140,142)(161,214,171,204)(162,213,172,203)(163,212,173,202)(164,211,174,201)(165,210,175,220)(166,209,176,219)(167,208,177,218)(168,207,178,217)(169,206,179,216)(170,205,180,215)(181,246,191,256)(182,245,192,255)(183,244,193,254)(184,243,194,253)(185,242,195,252)(186,241,196,251)(187,260,197,250)(188,259,198,249)(189,258,199,248)(190,257,200,247)(221,313,231,303)(222,312,232,302)(223,311,233,301)(224,310,234,320)(225,309,235,319)(226,308,236,318)(227,307,237,317)(228,306,238,316)(229,305,239,315)(230,304,240,314)>;
G:=Group( (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,196)(16,197)(17,198)(18,199)(19,200)(20,181)(21,217)(22,218)(23,219)(24,220)(25,201)(26,202)(27,203)(28,204)(29,205)(30,206)(31,207)(32,208)(33,209)(34,210)(35,211)(36,212)(37,213)(38,214)(39,215)(40,216)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,225)(62,226)(63,227)(64,228)(65,229)(66,230)(67,231)(68,232)(69,233)(70,234)(71,235)(72,236)(73,237)(74,238)(75,239)(76,240)(77,221)(78,222)(79,223)(80,224)(81,173)(82,174)(83,175)(84,176)(85,177)(86,178)(87,179)(88,180)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(241,269)(242,270)(243,271)(244,272)(245,273)(246,274)(247,275)(248,276)(249,277)(250,278)(251,279)(252,280)(253,261)(254,262)(255,263)(256,264)(257,265)(258,266)(259,267)(260,268)(281,301)(282,302)(283,303)(284,304)(285,305)(286,306)(287,307)(288,308)(289,309)(290,310)(291,311)(292,312)(293,313)(294,314)(295,315)(296,316)(297,317)(298,318)(299,319)(300,320), (1,214,48,285)(2,215,49,286)(3,216,50,287)(4,217,51,288)(5,218,52,289)(6,219,53,290)(7,220,54,291)(8,201,55,292)(9,202,56,293)(10,203,57,294)(11,204,58,295)(12,205,59,296)(13,206,60,297)(14,207,41,298)(15,208,42,299)(16,209,43,300)(17,210,44,281)(18,211,45,282)(19,212,46,283)(20,213,47,284)(21,160,308,185)(22,141,309,186)(23,142,310,187)(24,143,311,188)(25,144,312,189)(26,145,313,190)(27,146,314,191)(28,147,315,192)(29,148,316,193)(30,149,317,194)(31,150,318,195)(32,151,319,196)(33,152,320,197)(34,153,301,198)(35,154,302,199)(36,155,303,200)(37,156,304,181)(38,157,305,182)(39,158,306,183)(40,159,307,184)(61,279,177,107)(62,280,178,108)(63,261,179,109)(64,262,180,110)(65,263,161,111)(66,264,162,112)(67,265,163,113)(68,266,164,114)(69,267,165,115)(70,268,166,116)(71,269,167,117)(72,270,168,118)(73,271,169,119)(74,272,170,120)(75,273,171,101)(76,274,172,102)(77,275,173,103)(78,276,174,104)(79,277,175,105)(80,278,176,106)(81,137,221,247)(82,138,222,248)(83,139,223,249)(84,140,224,250)(85,121,225,251)(86,122,226,252)(87,123,227,253)(88,124,228,254)(89,125,229,255)(90,126,230,256)(91,127,231,257)(92,128,232,258)(93,129,233,259)(94,130,234,260)(95,131,235,241)(96,132,236,242)(97,133,237,243)(98,134,238,244)(99,135,239,245)(100,136,240,246), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,273,11,263)(2,272,12,262)(3,271,13,261)(4,270,14,280)(5,269,15,279)(6,268,16,278)(7,267,17,277)(8,266,18,276)(9,265,19,275)(10,264,20,274)(21,96,31,86)(22,95,32,85)(23,94,33,84)(24,93,34,83)(25,92,35,82)(26,91,36,81)(27,90,37,100)(28,89,38,99)(29,88,39,98)(30,87,40,97)(41,108,51,118)(42,107,52,117)(43,106,53,116)(44,105,54,115)(45,104,55,114)(46,103,56,113)(47,102,57,112)(48,101,58,111)(49,120,59,110)(50,119,60,109)(61,289,71,299)(62,288,72,298)(63,287,73,297)(64,286,74,296)(65,285,75,295)(66,284,76,294)(67,283,77,293)(68,282,78,292)(69,281,79,291)(70,300,80,290)(121,141,131,151)(122,160,132,150)(123,159,133,149)(124,158,134,148)(125,157,135,147)(126,156,136,146)(127,155,137,145)(128,154,138,144)(129,153,139,143)(130,152,140,142)(161,214,171,204)(162,213,172,203)(163,212,173,202)(164,211,174,201)(165,210,175,220)(166,209,176,219)(167,208,177,218)(168,207,178,217)(169,206,179,216)(170,205,180,215)(181,246,191,256)(182,245,192,255)(183,244,193,254)(184,243,194,253)(185,242,195,252)(186,241,196,251)(187,260,197,250)(188,259,198,249)(189,258,199,248)(190,257,200,247)(221,313,231,303)(222,312,232,302)(223,311,233,301)(224,310,234,320)(225,309,235,319)(226,308,236,318)(227,307,237,317)(228,306,238,316)(229,305,239,315)(230,304,240,314) );
G=PermutationGroup([(1,182),(2,183),(3,184),(4,185),(5,186),(6,187),(7,188),(8,189),(9,190),(10,191),(11,192),(12,193),(13,194),(14,195),(15,196),(16,197),(17,198),(18,199),(19,200),(20,181),(21,217),(22,218),(23,219),(24,220),(25,201),(26,202),(27,203),(28,204),(29,205),(30,206),(31,207),(32,208),(33,209),(34,210),(35,211),(36,212),(37,213),(38,214),(39,215),(40,216),(41,150),(42,151),(43,152),(44,153),(45,154),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,225),(62,226),(63,227),(64,228),(65,229),(66,230),(67,231),(68,232),(69,233),(70,234),(71,235),(72,236),(73,237),(74,238),(75,239),(76,240),(77,221),(78,222),(79,223),(80,224),(81,173),(82,174),(83,175),(84,176),(85,177),(86,178),(87,179),(88,180),(89,161),(90,162),(91,163),(92,164),(93,165),(94,166),(95,167),(96,168),(97,169),(98,170),(99,171),(100,172),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(241,269),(242,270),(243,271),(244,272),(245,273),(246,274),(247,275),(248,276),(249,277),(250,278),(251,279),(252,280),(253,261),(254,262),(255,263),(256,264),(257,265),(258,266),(259,267),(260,268),(281,301),(282,302),(283,303),(284,304),(285,305),(286,306),(287,307),(288,308),(289,309),(290,310),(291,311),(292,312),(293,313),(294,314),(295,315),(296,316),(297,317),(298,318),(299,319),(300,320)], [(1,214,48,285),(2,215,49,286),(3,216,50,287),(4,217,51,288),(5,218,52,289),(6,219,53,290),(7,220,54,291),(8,201,55,292),(9,202,56,293),(10,203,57,294),(11,204,58,295),(12,205,59,296),(13,206,60,297),(14,207,41,298),(15,208,42,299),(16,209,43,300),(17,210,44,281),(18,211,45,282),(19,212,46,283),(20,213,47,284),(21,160,308,185),(22,141,309,186),(23,142,310,187),(24,143,311,188),(25,144,312,189),(26,145,313,190),(27,146,314,191),(28,147,315,192),(29,148,316,193),(30,149,317,194),(31,150,318,195),(32,151,319,196),(33,152,320,197),(34,153,301,198),(35,154,302,199),(36,155,303,200),(37,156,304,181),(38,157,305,182),(39,158,306,183),(40,159,307,184),(61,279,177,107),(62,280,178,108),(63,261,179,109),(64,262,180,110),(65,263,161,111),(66,264,162,112),(67,265,163,113),(68,266,164,114),(69,267,165,115),(70,268,166,116),(71,269,167,117),(72,270,168,118),(73,271,169,119),(74,272,170,120),(75,273,171,101),(76,274,172,102),(77,275,173,103),(78,276,174,104),(79,277,175,105),(80,278,176,106),(81,137,221,247),(82,138,222,248),(83,139,223,249),(84,140,224,250),(85,121,225,251),(86,122,226,252),(87,123,227,253),(88,124,228,254),(89,125,229,255),(90,126,230,256),(91,127,231,257),(92,128,232,258),(93,129,233,259),(94,130,234,260),(95,131,235,241),(96,132,236,242),(97,133,237,243),(98,134,238,244),(99,135,239,245),(100,136,240,246)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260),(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300),(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)], [(1,273,11,263),(2,272,12,262),(3,271,13,261),(4,270,14,280),(5,269,15,279),(6,268,16,278),(7,267,17,277),(8,266,18,276),(9,265,19,275),(10,264,20,274),(21,96,31,86),(22,95,32,85),(23,94,33,84),(24,93,34,83),(25,92,35,82),(26,91,36,81),(27,90,37,100),(28,89,38,99),(29,88,39,98),(30,87,40,97),(41,108,51,118),(42,107,52,117),(43,106,53,116),(44,105,54,115),(45,104,55,114),(46,103,56,113),(47,102,57,112),(48,101,58,111),(49,120,59,110),(50,119,60,109),(61,289,71,299),(62,288,72,298),(63,287,73,297),(64,286,74,296),(65,285,75,295),(66,284,76,294),(67,283,77,293),(68,282,78,292),(69,281,79,291),(70,300,80,290),(121,141,131,151),(122,160,132,150),(123,159,133,149),(124,158,134,148),(125,157,135,147),(126,156,136,146),(127,155,137,145),(128,154,138,144),(129,153,139,143),(130,152,140,142),(161,214,171,204),(162,213,172,203),(163,212,173,202),(164,211,174,201),(165,210,175,220),(166,209,176,219),(167,208,177,218),(168,207,178,217),(169,206,179,216),(170,205,180,215),(181,246,191,256),(182,245,192,255),(183,244,193,254),(184,243,194,253),(185,242,195,252),(186,241,196,251),(187,260,197,250),(188,259,198,249),(189,258,199,248),(190,257,200,247),(221,313,231,303),(222,312,232,302),(223,311,233,301),(224,310,234,320),(225,309,235,319),(226,308,236,318),(227,307,237,317),(228,306,238,316),(229,305,239,315),(230,304,240,314)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 30 |
| 0 | 0 | 11 | 27 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 38 |
| 0 | 0 | 17 | 38 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,32,11,0,0,30,27],[40,0,0,0,0,1,0,0,0,0,3,17,0,0,38,38] >;
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D5 | C4○D4 | D10 | D10 | Dic10 | C4×D5 | C4○D20 |
| kernel | C2×C4×Dic10 | C4×Dic10 | C2×C4×Dic5 | C2×C10.D4 | C2×C4⋊Dic5 | C2×C4×C20 | C22×Dic10 | C2×Dic10 | C2×C20 | C2×C42 | C2×C10 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 16 | 4 | 2 | 4 | 8 | 6 | 16 | 16 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_4\times Dic_{10} % in TeX
G:=Group("C2xC4xDic10"); // GroupNames label
G:=SmallGroup(320,1139);
// by ID
G=gap.SmallGroup(320,1139);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=c^10,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀